Somewhat surprisingly, the answer is no: one guest is OK, but additional kids are not. From customer service:
Please be advised that you only have one guest allowance included when you access our lounges using your [Krisflyer Gold] membership card. However, you can purchase 2 Single Entry passes to the lounge for your 2 children for $65 (per person). It is payable at the lounge reception by a credit card and can be used for a maximum of 2 hours before departure time. This is available for domestic lounges only and does not include partner or international lounges and is subject to space availability.
However, we tried our luck anyway and were let in -- although the lounge staff did note that technically kids are not included, and we were getting a free pass because the lounge was not too full.
Also, for what it's worth, the lounge breakfast is nice enough, but you'd need feed your kids an awful lot of bacon egg muffins for that $65 to be worth it. And you can't even add some sparkling wine to your orange juice, since they don't open the bar until 11am!